Consider the block matrix given by $$M = \begin{bmatrix} A_{11}&A_{12}&0\\ A_{12}&A_{22}&A_{23}\\ 0&A_{23}&A_{33}\end{bmatrix}$$
What conditions should I impose on each matrix $A_{ij}$ to ensure that M is positive definite? Each matrix $A_{ij}$ is symmetric. I would like not to use a Schur complement-like condition to avoid deal with inverses. Also, could someone provide a good reference on this subject?
Thank you
Based on the post by user1551, I came up with the following proof.
If each matrix $A_{ij}$ is such that the block submatrices
\begin{equation*} M_1=\begin{bmatrix} A_{11}&A_{12}\\ A_{12}&A_{22}/2 \end{bmatrix}, M_2=\begin{bmatrix} A_{22}/2&A_{23}\\ A_{23}&A_{33} \end{bmatrix} \end{equation*} is positive definite, then the matrix $M$ is positive definite.
Proof. By assumption $M_1$ and $M_2$ are positive definite. Consequently, for vectors $(x,y)$ and $(y,z)$ the following inequalities hold \begin{align*} x^\top A_{11}x + x^\top A_{12}y + y^\top A_{12} x + y^\top \dfrac{A_{22}}{2} y&>0\\ y^\top \dfrac{A_{22}}{2}y + y^\top A_{23}z + z^\top A_{23} y + z^\top A_{33} z&>0 \end{align*} Summing both equations yields \begin{equation} x^\top A_{11}x + x^\top A_{12}y + y^\top A_{12} x + y^\top A_{22}y+y^\top A_{23}z + z^\top A_{23} y + z^\top A_{33} z>0. \end{equation} This inequality implies that the matrix $M$ is positive definite.